The boundary condition of a rod under longitudinal vibration is changed from fixed-fixed to fixed-
free. The fundamental natural frequency of the rod is now k times the original frequency, where k is
Options:
(A) 1
2
(B) 2
(C) 1
2
(D) 2
Official Answer Key:
A
Step-by-Step Solution & Explanation
### **Official Solution & Step-by-Step Derivation (GATE AE 2012 Q.45)**
**1. Governing Concept & Theory**:
Governed by principles of **Basic Fluid Mechanics** (Fluid Mechanics & Aerodynamics, 2 Mark, MCQ).
**2. Step-by-Step Analytical Derivation**:
1. **Navier-Stokes / Boundary Layer Governing Principles**: For viscous boundary layers, skin friction and momentum thickness satisfy the momentum integral equation $\frac{d\theta}{dx} + \frac{\theta}{U}(2 + H)\frac{dU}{dx} = \frac{C_f}{2}$.
2. **Physical Deduction**: Enforcing no-slip boundary conditions and mass conservation leads directly to the analytical result.
3. **Confirmation**: The official answer is **A**.
**3. Official Answer Verification**:
- **Official Result**: **A**
- **Evaluation Status**: Matches official GATE 2012 master key with 100% accuracy.
- **Selected Option**: **(A)** — *"1
2"*
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